A218009 Binomial transform of A212831(n).

0, 1, 4, 12, 30, 70, 160, 364, 824, 1848, 4096, 8976, 19488, 42016, 90112, 192448, 409472, 868224, 1835008, 3866880, 8126976, 17039872, 35651584, 74447872, 155187200, 322959360, 671088640, 1392513024, 2885689344, 5972697088, 12348030976

Offset

0,3

Comments

Companion to A217988.

Considering a(n+1) - 2*a(n) = 1,2,4,6,10,20,44,96,200,... = b(n), is

b(n+3) - 2*b(n+2) = -2,-2,0,4,8,8,0,-16,-32,-32,0,... = -A009545(n+2).

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (6,-14,16,-8).

Formula

G.f.: x*(1-2*x+2*x^2-2*x^3+2*x^4)/((1-2*x)^2*(1-2*x+2*x^2)). - Bruno Berselli, Oct 18 2012

a(n) = 2^(n-3)*(3*n+2)+(1+i)^(n-4)+(1-i)^(n-4), where i=sqrt(-1) and n>1, with a(0)=0, a(1)=1. - Bruno Berselli, Oct 18 2012

a(n) = 6*a(n-1) -14*a(n-2) +16*a(n-3) -8*a(n-4).

Mathematica

a[n_] := Sum[(1/4)*Binomial[n, k]*((-(1 + (-1)^k))*(-1 + (-1)^Floor[k/2]) - (-3 + (-1)^k)*k), {k, 0, n}]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Oct 18 2012 *)
CoefficientList[Series[x*(1 - 2*x + 2*x^2 - 2*x^3 + 2*x^4)/((1 - 2*x)^2*(1 - 2*x + 2*x^2)), {x, 0, 40}], x]; // Vincenzo Librandi, Dec 15 2012

Prog

(Magma) I:=[0, 1, 4, 12, 30, 70]; [n le 6 select I[n] else 6*Self(n-1) - 14*Self(n-2) + 16*Self(n-3) - 8*Self(n-4): n in [1..40]]; // Vincenzo Librandi, Dec 15 2012

Crossrefs

Cf. A214282, A214283.

Sequence in context: A036388 A100691 A000298 * A249914 A006802 A068055

Adjacent sequences: A218006 A218007 A218008 * A218010 A218011 A218012

Keyword

nonn,easy

Author

Paul Curtz, Oct 18 2012

Status

approved